ERG SES H 07, Sciences, Mathematics and Education
It is important for pre-service teachers to have content and pedagogical knowledge in the subject matter to teach mathematics effectively and overcome problematic situations that may arise in classrooms (Shulman, 1986; Ball, Thames, & Phelps, 2008; Morris, Hiebert, & Spitzer, 2009). It is considered that "definitions of mathematical concepts, the underlying structures of the definitions and the process of defining are some of the fundamental components of the subject matter knowledge of teachers of mathematics" (Zazkis & Leikin, 2008, p. 133). Concept definition helps educational researchers to examine defining process and to understand how the mathematical concepts were shaped in minds (Edwards & Ward, 2008). Students’ incomplete understanding or lack of formal knowledge might be revealed through defining (Peled & Hershkovitz, 1999; Fischbein, Jehiam, & Cohen, 1995; Zazkis & Leikin, 2008). Teachers are one of the important components in students’ learning process as their understanding of mathematics is central for effective teaching (Feueborn, Chinn, & Morlan, 2009). Some mathematical contents which are really hard for elementary students may be starting point to understand pre-service teachers’ knowledge and to complete their necessities up to their graduation. One of the fundamental but not an easy topic for pre-service teachers to construct in their minds is integers (Smith, 2002; Steiner, 2009; Akyüz, Stephan, & Dixon, 2012).
Accordingly, the aim of the study is to examine the nature of pre-service elementary mathematics teachers’ subject matter knowledge regarding descriptions of integers and quoted definitions of integers. Leikin and Zazkis (2010) suggested criteria for the analysis of teacher-generated examples of definitions. According to the framework, definitions can be evaluated in terms of the following criteria: accessibility, correctness (appropriateness), richness, generality/concreteness. In this study, a correctness criterion is focused on to examine prospective teachers understanding about the integer concept. "Correctness refers to the properties of the examples generated" (Leikin & Zazkis, 2010, p.456). It is examined as appropriate and inappropriate example statements.
Inappropriate statements: "Inappropriate examples of definitions of mathematical concepts are examples which are lacked either necessary or sufficient conditions, so that they represented mostly specific instances of the concepts" (Leikin & Zazkis, 2010, p.459).
Appropriate statements: Appropriate examples of definitions of mathematical concepts are identified in two: "(1) appropriate rigorous examples of definitions are examples which include necessary and sufficient conditions of the defined concept as well as accurate mathematical terminology and symbols, and are usually minimal and (2) appropriate but not rigorous examples of definitions are examples which usually omit some constraint or use imprecise terminology because of a lack of attentiveness on the part of the PMT or a lack of rigor in the mathematical language in the usual mathematics classroom." (Leikin & Zazkis, 2010, p.457).
In this study, pre-service teachers’ conceptions of integers were investigated based on the following research questions: How appropriate are the integer definitions made by pre-service elementary mathematics teachers? What are the interpretations made by pre-service teachers regarding quoted definitions of integers?
Akyüz, D., Stephan, M., & Dixon, J. K. (2012). The role of the teacher in supporting imagery in understanding integers. Education and Science, 37(163), 268-282. Ball, D., L. (2000). Bridging practices intertwining content and pedagogy in teaching and learning to teach. Journal of Teacher Education, 51(3), 241 – 247. Feueborn, L. L., Chinn, D., & Morlan, G. (2009). Improving mathematics teachers’ content knowledge via brief in-service: a US case study. Professional Development in Education, 35(4), 531-545. Fischbein, E., Jehiam, R., & Cohen, D. (1995). The concept of irrational numbers in high-school students and prospective teachers. Educational Studies in Mathematics, 29(1), 29-44. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: mathematical knowing and teaching. American Educational Research Journal, 27(1), 29-63. Leikin, R., & Zaskis, R. (2010). On the content-dependence of prospective teachers’ knowledge: a case of exemplifying definitions. International Journal of Mathematical Education in Science and Technology, 41(4), 451–466. Morris, A. K., Hiebert, J., & Spitzer, S. M. (2009). Mathematical knowledge for teaching in planning and evaluating instruction: What can pre-service teachers learn? Journal for Research in Mathematics Education, 40(5), 491-529. Peled, I. (2010). Difficulties in know ledge integration: revisiting Zeno's paradox with irrational numbers. International Journal of Mathematical Education in Science and Technology, 30(1), 39-46 Shield, M. (2004). Formal definitions in mathematics. Australian Mathematics Teacher, 60 (4), 25 – 28. Shulman, L. (1986). Those who understand knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Smith, T. (2002). Using manipulatives with fractions, decimals, integers, and algebra: Guide for the intermediate teacher. Unpublished Master’s Thesis, Memorial University of Newfounland. Steiner, C. J. (2009). A Study of pre-service elementary teachers’ conceptual understanding of integers. Unpublished doctoral dissertation, Kent State University College and Graduate School of Education. Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: a case of a square. Educational Studies of Mathematics, 69, 131–148. Vinner, S. (1991). The Role of Definitions in Teaching and Learning in Tall, D.(ed.) Advanced Mathematical Thinking, (pp. 65-81). Dordrecht: Kluwer. Van Dormolen, J., & Zaslavsky, O. (2003).The many facets of a definition: The case of periodicity. Journal of Mathematical Behavior, 22, 91–106. Yin, R. K. (2003). Case study research design and methods (3rd ed.). Sage Publications, Inc: USA.
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